If $A$, $B$ and $A + B$ are invertible, then $A^{-1} + B^{-1}$ is invertible.

Question

$A$ and $B$ are $n \times n$ square matrices. $A$, $B$ and $A + B$ are invertible.

Show that $A^{-1} + B^{-1}$ is invertible.

This may be an easy problem but in my book there are no solutions to the exercises. And I could not manage to solve this part of the exercise for much too long.

Answer 1

It's because $A^{-1}+B^{-1}=(I+B^{-1}A)A^{-1}=B^{-1}(B+A)A^{-1}$